Let $X$ be a complex algebraic variety and $G$ a complex reductive group acting on $X$. Let $L$ be a linearization of this action, i.e. a line bundle with a linear action of $G$ covering that of $X$. Recall that a point $x\in X$ is semistable if there is an invariant section $\sigma\in H^0(X,L^r)^G$ for some $r\ge0$ such that $\sigma(x)\neq 0$.
When $X$ is affine there is a useful criterion for semistability (A. D. King, Quart. J. Math. Oxford (2), 45 (1994), 515-530):
Proposition. A point $x\in X$ is semistable if and only if for any non-zero lift $\hat{x}\in L$ of $x$ the orbit closure $\overline{G\cdot\hat{x}}$ is disjoint from the zero-section.
Is this criterion also valid in the general case?
